Transitive Anosov flows and Axiom-A diffeomorphisms

Abstract

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyper-bolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessity of suitable conditions on the cocycle.

Upper Bound on the Density of Ruelle Resonances for Anosov Flows

“…Notice that in this case, the geodesic flow is Anosov on the unit cotangent bundle X = T * 1 M. • The global hyperbolic structure of Anosov flows or Anosov diffeomorphisms is a very strong geometric property, so that manifolds carrying such dynamics satisfy strong topological conditions and the list of known examples is not so long. See [7] for a detailed discussion and references on that question.…”

On the Three-Legged Accessibility Property

“…This generalizes Burns, Pugh and Wilkinson [5] result about the accessibility of Anosov flows. Also, as an application of Theorem 1.1 we obtain a generalization of the main result of Bonatti and Guelman [1] about the approximation of time-one maps of Anosov flows by Axiom A diffeomorphisms.…”

“…We also give some applications to skew products (Section 5) and to Anosov diffeomorphisms (Section 6). In the case of skew products we 1 The authors jointly with Federico Rodriguez Hertz already had a proof of this result more than ten years ago.…”

Uniqueness of minimal unstable lamination for discretized Anosov flows

“…We point out that our proof relies on a different approach. The main inspiration for the present work comes from [BG09] where it was shown that every Axiom A discretized Anosov flow as in the hypothesis of Theorem A admits a unique attractor. By generalizing the arguments in [BG09] (see also [G02]) we are able to remove the Axiom A hypothesis and to obtain not only uniqueness of quasi-attractor but also of minimal unstable lamination.…”